PACKAGES AND POLYGONS

GOALS

1. recognize and identify geometrical shapes and structures in real objects and in representations

2. understand and use Euler’s formula

3. use the relationship between angles and turns to solve problems

4. recognize and identify properties of regular polygons and polyhedra

5. construct geometric models

6. draw two- and three-dimensional figures

7. make connections between different views of geometric solids

8. develop efficient counting strategies, involving geometric solids, that can be generalized

9. reason about the structure of the Platonic solids

10. develop spatial visualization skills

11. solve problems involving geometric solids

SECTION SUMMARIES

SECTION A: PACKAGES

1. Pyramids and prisms have edges that are straight lines; pyramids and cones come to a single point; a sphere does not have any straight edges; cubes and prisms have flat sides, but a cylinder has a round side as well.

2. The word “truncate” means to shorten by cutting off ( as in a truncated cone or pyramid – chopped off ).

SECTION B: NETS

1. A net is a flat pattern that forms a three-dimensional shape when folded.

2. All of the faces of a shape can be seen in a net. You can also make different nets for the same shape.

SECTION C: BAR MODELS

1. Drawing a net and folding the sides together is one way to make a model of a 3-D shape. Another way is to make a bar modle. The bars are the edges of the shape. The point at which two or more bars meet is called a vertex.

2. A structure may be made more stable by creating more triangles within it. Sometimes the new triangles are parts of the original faces and at other times, they are not.

SECTION D: POLYGONS

1. Polygons are two-dimensional shapes with three or more angles, and are named according to the numer of sides or angles they have. In order - triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, undecagon, and dodecagon.

2. A polygon having n sides (unspecified) is called an n-gon. ie a 13-gon has 13 sides.

3. A regular polygon has equal sides and equal angles. Knowing how many turns you would have to make to move around a regular polygon, you can find the measure for each inside angle. A complete turn = 360degrees.

SECTION E: PLATONIC SOLIDS

1. There are only five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) and they are the only polyhedra that are regular. There are many relationships among the edges, faces, and vertices of the Platonic solids.

SECTION F: EULER’S FORMULA

1. The relationship between the number of faces, vertices, and edges of any polyhedron is expressed in Euler’s

formula, F + V – E = 2.

SECTION G: SEMI-REGULAR POLYHEDRA

1. Semi-regular polyhedra have at least two different regular polygons as faces. These solids can be made by

cutting pieces off regular polyhedra. For example, if you cut pieces off the corners of an icosahedron, you get a semi-regular polyhedron made of pentagons and hexagons.